Transformers Learn to Achieve Second-Order Convergence Rates for In-Context Linear Regression

TL;DR

This paper reveals that Transformers perform in-context linear regression by approximating second-order optimization methods, specifically Newton's method, achieving faster convergence than first-order methods like Gradient Descent.

Contribution

The paper demonstrates that Transformers learn to implement second-order optimization, matching Newton's method's convergence rate, and provides theoretical proof of their ability to simulate multiple iterations with few layers.

Findings

Transformers' predictions closely match Newton's method iterations.

Transformers converge exponentially faster than Gradient Descent.

They can handle ill-conditioned data where GD struggles.

Abstract

Transformers excel at in-context learning (ICL) -- learning from demonstrations without parameter updates -- but how they do so remains a mystery. Recent work suggests that Transformers may internally run Gradient Descent (GD), a first-order optimization method, to perform ICL. In this paper, we instead demonstrate that Transformers learn to approximate second-order optimization methods for ICL. For in-context linear regression, Transformers share a similar convergence rate as Iterative Newton's Method, both exponentially faster than GD. Empirically, predictions from successive Transformer layers closely match different iterations of Newton's Method linearly, with each middle layer roughly computing 3 iterations; thus, Transformers and Newton's method converge at roughly the same rate. In contrast, Gradient Descent converges exponentially more slowly. We also show that Transformers can learn in-context on ill-conditioned data, a setting where Gradient Descent struggles but Iterative Newton succeeds. Finally, to corroborate our empirical findings, we prove that Transformers can implement $k$ iterations of Newton's method with $k + \mathcal{O}(1)$ layers.